Dallas Cowboys (16)
the pac-10
The SEC has a total of 6. Four more than any other conference.
NFC East
Probably Ohio State, USC, Oklahoma, North Carolina, or Notre Dame.
The Cleveland Browns have played in a total of 10 AFC Championship Games. Their appearances were primarily during the 1980s, with their last appearance occurring in the 1989 season. The Browns have yet to win an AFC Championship, which has contributed to their status as one of the teams with the most conference championship game losses.
The Minneapolis/Los Angeles Lakers have appeared in the most championship series -- 31.
The Big 12 is a class A conference of the National College Athletic Association. Information on the Big 12 Championship can generally be found on sports oriented websites and television channels and in general news media as well as sports magazines. The official site of the conference, Big12Sports, is probably the most reliable source for timely information.
The losers of the Super Bowl do receive a championship ring, but it is for whichever conference they are from. There are rules regarding it's design, though. For example, it can only cost (at most) half as much as the winners' rings cost.
Since the founding of the franchise in 1989 as a result of the expansion of the NBA, the Orlando Magic have had the most successful run in terms of winning percentage of the four teams in the expansion, but has never won a championship. They have won 2 conference titles (1995 and 2009), but never won the championship.
The Montreal Canadiens with 24 Championship Victories.
The Pac 10 with 15 Men's NCAA championships. UCLA - 11 Oregon - 1 Stanford - 1 California - 1 Arizona - 1
Through the 2009 season, that is the New York Giants with 18 NFL championship game appearances (4 Super Bowls and 14 NFL championship games prior to the beginning of the Super Bowl).